Preheat oven to 375° F.
Cream butter and sugar until smooth. Beat in egg and vanilla. Slowly add flour, baking soda, baking powder, and cardamom, and mix to combine. Add ginger and apricots.
Using a cookie scoop or lightly greased hands, make small scoops or balls of dough onto baking pans (non-stick or use silpat baking sheets).
Bake 8-10 minutes or until golden. Let stand on cooking sheet at least 5 minutes before moving to cooling racks.
For glaze, mix all ingredients, and add more water if needed to make a drizzly substance. Use a fork to quickly add a small amount of glaze to each cookie.